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Abstract

At present, traditional analysis methods sometimes show incompatibility when analyzing the degree of freedom (DOF) of parallel mechanisms. In this research, a new DOF analysis method (MV-DOF method) based on primary-ancillary motion theory is developed. The 4PPRRR series-parallel mechanism is taken as an example, the analysis results by using the MV-DOF method are compared with those of traditional inverse helix theory and modified Grübler-Kutzbach formula, respectively. The comparison shows that this MV-DOF method can always analyze the DOF of the 4PPRRR series-parallel mechanism accurately, while the other two traditional methods show inapplicability sometimes. During the analysis, an important rule is also found that the ancillary motion is equivalent to local constraint helix motion, but it is not always true in turn. Based on this MV-DOF method, a DOF cutting analysis way is also suggested. Any output point on a complex mechanism can be taken as a cutting point, along which the mechanism can be cut into two independent sub-mechanisms. The output DOF of each sub-mechanism can be analyzed separately by using the MV-DOF method, then the intersection can be obtained as the DOF of the whole mechanism at this output point. This research provides a new and effective analytical way for DOF analysis of parallel mechanisms.

Keywords

Parallel mechanism degree of freedom primary-ancillary motion MV-DOF cutting point

Introduction

Grübler-Kutzbach formula^1–4 (abbreviated as the G-K formula), based on basic arithmetic operations, has been widely used for determining the degree of freedom (DOF) of the mechanism, which mainly includes two forms of plane mechanism and space mechanism. In recent years, parallel mechanisms have attracted wide interest and have been proposed.^5–7 However, some parallel mechanism was found with exceptions to the above formulas with the developments of mechanism. Through systematic analysis, Gogu⁸ pointed out that the traditional formula for fast calculation of DOF is not suitable for some modern parallel mechanisms. Isaksson et al.⁹ reported the inapplicability of this method for a class of 3-DOF parallel manipulators with axis-symmetric arm systems. Many researchers have attempted to look for new DOF analysis methods.^10–13 During this period, a DOF analysis method based on the inverse helix theory was developed and applied well,^14–16 which avoided the nonlinear problem caused by using the unified coordinate system to analyze the different branches of the parallel mechanism, and greatly simplified the analysis process. On this basis, Huang et al.¹⁷ further developed a general DOF formula based on the inverse helix theory, referred to as the modified G-K formula, which has been successfully used for DOF analysis of many complex mechanisms that cannot be solved by traditional G-K formulas. However, the above studies mainly focused on the numerical calculation of the DOF, and they were not based on the mathematical representation of the motion dimension of the object's motion trajectory in space to analyze the composition, transformation, and synthesis of the basic elements of its input and output. Moreover, none of them distinguished the output primary motion and the output ancillary motion, and reflected the relationship between the input motion and output motion, thus affecting the analysis accuracy.

In this research, we clearly define the primary motion and ancillary motion of the parallel mechanism for more accurate DOF analysis. It is worth noting that the ancillary motion we refer to here is different from the parasitic motion in the parallel mechanism. As we all know, parasitic motion often occurs in the parallel mechanism. It is a lower amplitude motion that occurs on non-independent DOF.^18,19 Although the value is small, it has a great influence on the application of parallel mechanisms. At present, scholars’ research on parasitic motion is mostly focused on [PP]S and 2R1 T types of parallel mechanisms. The main purpose of this research was to find mechanism configurations without parasitic motion or control methods to reduce parasitic motion.^20–23 However, here we clearly distinguish the output motion into primary motion and ancillary motion. The primary motion here refers to the output motion element of the same form as the input motion, and it is obtained by the input motion element through one-to-one transfer mapping. The ancillary motion defined in this research has a different form from the input motion, which is obtained by a comprehensive correlation of two or more specifically combined input motion elements, and is caused by the non-independent transfer mapping of input motion. The purpose of this distinction is not to reduce ancillary motion but to better analyze the relationship between the input motion and the output motion, and more accurately obtain the DOF of the parallel mechanism.

In this paper, based on the primary-ancillary motion theory and its six-element characteristic numerical method, a new analysis method (MV-DOF method) is constructed. Taking the 4PPRRR series-parallel mechanism as an example, the DOF analysis results by using this proposed method are compared with those of traditional inverse helix theory and modified G-K formula respectively. The comparison results show that, for a complete 4PPRRR series parallel mechanism, these two traditional methods can accurately obtain the DOF, but there are obvious errors when some motion pairs with the same spatial configuration are added or reduced in branch chains of the mechanism. However, the MV-DOF method can always correspond to the transformation evolution process of the parallel mechanism, and can accurately calculate the DOF regardless of the change of the branch chains.

Compared with traditional DOF analysis ways, this MV-DOF analysis method proposed here has the following characteristics: the primary motion and ancillary motion of the mechanism are explicitly distinguished, the DOF analysis process is based on the mapping relationship between the input and output primary-ancillary motion, this method shows higher accuracy and applicability in analyzing the 4PPRRR series-parallel mechanism than traditional ways. In addition, an important rule is also revealed during the analysis that the ancillary motion is equivalent to local constraint helix motion, but local constraint helix motion is not exactly the ancillary motion.

Primary-ancillary motion

Motion is introduced into the mechanism through the input motion joints in the mechanism, and is synthetically transformed into the motion of the output joints or members by the combined joint action of multiple joints. The basic motion elements of the input joints include linear motion elements and rotational motion elements.

First, when the branch chain or mechanism has a combination of PP motion pairs, there are two cases of perpendicular and non-perpendicular spatial axes of the two motion pairs. When their spatial axes are perpendicular, the corresponding input motion trajectory in space is integrated into a spatial plane rectangular domain. The input motion corresponds to two independent orthogonal linear motions in the space coordinate system. Based on the spatial physical-mathematical representation, the output motion contains two independent orthogonal linear motions in the same direction as the axes of the input linear motion and also contains one non-independent rotational motion that is orthogonal to the axes of the input linear motion and is obtained by the combined action of the two input linear motion elements.

Second, when the branch chain or mechanism has a combination of RR motion pairs, there are also two cases of parallel and non-parallel spatial axes of the two motion pairs. When their spatial axes are parallel, the corresponding input motion trajectory in space is integrated into a spatial plane circular domain, the input motion corresponds to two independent parallel axes of rotation motion in the space coordinate system. The output motion contains one independent rotational motion in the same direction as the axis of the input linear motion and also contains two non-independent linear motions that are orthogonal to the axis of the input rotational motion and are obtained by the combined action of the two input rotational motion elements.

Third, when the branch chain or mechanism has a combination of RP motion pairs, there are also two cases of parallel and non-parallel spatial axes of the two motion pairs. When their spatial axes are perpendicular, the corresponding input motion trajectory in space is integrated into a spatial plane circular ring domain, and the input motion corresponds to a rotation motion and a linear motion whose axes are perpendicular to each other. The output motion consists of one linear motion in the same direction as the axis of the input linear motion and one rotation motion in the same direction as the axis of the input rotation motion, and also one non-independent linear motion which is orthogonal to the input rotational motion and the input linear motion, and obtained by the combined action of the two input motion elements.

The output independent motion obtained in the above three cases is defined as the primary motion and the output non-independent motion obtained is defined as the ancillary motion. ${P \circ}_{i}, {P \circ}_{j}, and {P \circ}_{k}$ are defined as the three basic units of linear motion along the spatial orthogonal axis i, j, k directions, and ${R \circ}_{i}, {R \circ}_{j}, and {R \circ}_{k}$ are three basic units of rotational motion around the spatial orthogonal axis i, j, k directions, respectively, as shown in Table 1. It can be seen from Table 1, the rotational motion ${R \circ}_{k}$ obtained by the combined action of the two linear input motions PP is called the output ancillary motion, and the output motions ${P \circ}_{i}$ and ${P \circ}_{j}$ obtained by the transfer action of original linear input motions are defined as the output primary motions. The linear motion ${P \circ}_{j}$ and ${P \circ}_{k}$ obtained by the combined action of two rotational input motions RR is called the output ancillary motion. The output motion ${R \circ}_{k}$ obtained by the transfer action of the original rotational input motion is defined as the output primary motion. Similarly, the linear motion ${P \circ}_{k}$ is called the output ancillary motion, and the output motions ${R \circ}_{i}$ and ${P \circ}_{j}$ are defined as the output primary motions.

Table 1.

Terms and concepts related to primary-ancillary motion.

Motion pairs	Spatial relationship of axes in motion pairs	Output primary motion	Output ancillary motion
$PP$	P is perpendicular to P	${P \circ}_{i}, {P \circ}_{j}$	${R \circ}_{k}$
$PP$	P is not perpendicular to P	${P \circ}_{i} / {P \circ}_{j}$	${P \circ}_{j} / {P \circ}_{i}, {R \circ}_{k}$
$RR$	R is parallel to R	${R \circ}_{i}$	${P \circ}_{j}, {P \circ}_{k}$
$RR$	R is not parallel to R	${R \circ}_{i}, {R \circ}_{j}$	Φ
$RP$	R is perpendicular to P	${R \circ}_{i}, {P \circ}_{j}$	${P \circ}_{k}$
$RP$	R is not perpendicular to P	${R \circ}_{i}, {P \circ}_{j}$	Φ

In order to describe the input motion, output primary motion, and output ancillary motion of the branch chain or mechanism scientifically and effectively, and to lay the foundation for subsequent mathematical operations and realization of programmed operations, the following definitions and relevant operation rules are given.

Operator symbols. The superposition or synthesis between and within motion pairs is represented by the generalized plus “ $\oplus$ .” The union operation between different types of output motion (output primary motion, output ancillary motion) within a certain branch chain is represented by the symbol “ $\cup$ .” The intersection operation between the output motions and the corresponding motion types of a branch chain is represented by the symbol “ $\cap$ .”

Six-element matrix description of motion. The input and output primary linear motions along the spatial axes x, y, and z are represented by the symbol $\bar{x}, \bar{y,} and \bar{z}$ . The input and output primary rotational motions around the spatial axes x, y, and z are denoted by the symbol $\hat{x}$ , $\hat{y}$ , and $\hat{z}$ , the corresponding matrix is expressed as $[\begin{matrix} \bar{x} & \bar{y} & \bar{z} & \hat{x} & \hat{y} & \hat{z} \end{matrix}]$ , and the numerical motion matrix is expressed as $[\begin{matrix} 1 & 2 & 3 & 4 & 5 & 6 \end{matrix}]$ . The first-order output ancillary linear motions along the spatial axes x, y, and z are denoted by the symbol ${\bar{x}}^{'}, {\bar{y}}^{'}, and {\bar{z}}^{'}$ . The first-order output ancillary rotational motion around the spatial axes x, y, and z are denoted by the symbol ${\hat{x}}^{'}$ , ${\hat{y}}^{'}$ , and ${\hat{z}}^{'}$ , e corresponding matrix is expressed as $[\begin{matrix} {\bar{x}}^{'} & {\bar{y}}^{'} & {\bar{z}}^{'} & {\hat{x}}^{'} & {\hat{y}}^{'} & {\hat{z}}^{'} \end{matrix}]$ , and the numerical motion matrix is expressed as $[\begin{matrix} 01 & 02 & 03 & 04 & 05 & 06 \end{matrix}]$ .

Comparison of different DOF analysis methods for parallel mechanism

The inverse helix theory and the modified G-K formula are commonly used to analyze the DOF of parallel mechanisms. The inverse helix theory is based on the basic helix theory. By establishing the motion helix system of the mechanism, the inverse helix system can be calculated and the DOF of the mechanism can be analyzed. The modified G-K formula is also based on the basic helix theory, but the difference is that the over constraint and the general constraint in parallel mechanism are redefined by inverse helix theory, so as to determine the constraint of the mechanism more conveniently. Different from these two traditional methods, the MV-DOF method proposed in this paper clearly distinguishes the output primary motion and output ancillary motion of the mechanism, and realizes the accurate DOF analysis based on the mapping relationship between the input motion and the output primary and ancillary motions.

In this article, a 4PPRRR series parallel mechanism is adopted as an example, as shown in Figure 1. The DOF analysis results by using the MV-DOF method are compared with the two traditional methods above.

Figure 1.

Sketch of a 4PPRRR parallel mechanism.

Here, the PRRR-3PPRRR (delete the motion pair $A_{1}$ in Figure 1), RRR-PRRR-2PPRRR (delete the motion pairs $A_{1}, A_{2}, and B_{1}$ in Figure 1), 2RRR-PRRR-PPRRR (delete the motion pairs $A_{1}, A_{2}, B_{1}, B_{2}, and C_{1}$ in Figure 1), 2RRR-2PRRR (delete the motion pairs $A_{1}, A_{2}, B_{1}, B_{2}, C_{1}, and D_{1}$ in Figure 1), and 4RRR (delete the motion pairs $A_{1}, A_{2}, B_{1}, B_{2}, C_{1}, C_{2}, D_{1}, and D_{2}$ in Figure 1) are analyzed by using inverse helix theory, modified G-K formula and the MV-DOF method, respectively.

For the 4PPRRR parallel mechanism, based on the helix theory, the motion helix system of branch chain 1 can be obtained as follows:

{\begin{matrix} \begin{matrix} \begin{matrix} $_{11} = (0 0 0; 0 1 0) \\ $_{12} = (0 0 0; 0 0 1) \\ $_{13} = (0 0 1; a_{1} 0 c_{1}) \end{matrix} \\ $_{14} = (0 0 1; a_{2} 0 c_{2}) \end{matrix} \\ $_{15} = (0 1 0; a_{3} 0 c_{3}) \end{matrix} (1)

(1)

By calculating the inverse helix based on this helix system, the constraint helix system (primary inverse helix) of branch chain 1 can be obtained as follows:

$_{1}^{r} = (0 0 0; 1 0 0)

(2)

Similarly, the motion helical system of branch chain 2 can be obtained as follows:

{\begin{matrix} $_{21} = (0 0 0; 1 0 0) \\ $_{22} = (0 0 0; 0 0 1) \\ $_{23} = (0 0 1; 0 b_{1} c_{13}) \\ $_{24} = (0 0 1; 0 b_{2} c_{4}) \\ $_{25} = (1 0 0; 0 b_{3} c_{5}) \end{matrix}

(3)

The constraint helix system (primary inverse helix) of branch chain 2 can be obtained as follows:

$_{2}^{r} = (0 0 0; 0 1 0)

(4)

Because of the symmetric structural form of the mechanism,

$_{1}^{r} = $_{3}^{r}, $_{2}^{r} = $_{4}^{r}

, and the local constraint, the helix is S3; so there is

μ = 2

. From the primary inverse helix system, the secondary inverse helix system can be obtained as follows:

{\begin{matrix} $_{m 1}^{r r} = (0 0 1; 0 0 0) \\ $_{m 2}^{r r} = (0 0 0; 1 0 0) \\ $_{m 3}^{r r} = (0 0 0; 0 1 0) \\ $_{m 4}^{r r} = (0 0 0; 0 0 1) \end{matrix}

(5)

By using the modified G-K formula, the DOF of the parallel mechanism can be given as follows:

\begin{aligned} M = & 6 - d (n - g - 1) + \sum_{i = 1}^{g} f_{i} + μ \\ = & 6 - 0 (18 - 20 - 1) + 20 + 2 = 4 \end{aligned}

(6)

where M represents the DOF of the mechanism, d represents the order of the mechanism, n represents the number of components, g represents the number of motion pairs,

f_{i}

represents the DOF of the motion pair i, and

μ

indicates the number of parallel redundancy constraints.

Analysis of the four branches of the mechanism based on the MV-DOF method is given as follows:

{\begin{matrix} O b_{1} = [\begin{matrix} 01 & 2 \oplus 02 & 3 \end{matrix} \begin{matrix} 04 & 5 & 6 \end{matrix}] \\ O b_{2} = [1 \begin{matrix} \oplus 01 & 02 & 3 \end{matrix} \begin{matrix} 4 & 05 & 6 \end{matrix}] \\ O b_{3} = [\begin{matrix} 01 & 2 \oplus 02 & 3 \end{matrix} \begin{matrix} 04 & 5 & 6 \end{matrix}] \\ O b_{4} = [1 \begin{matrix} \oplus 01 & 02 & 3 \end{matrix} \begin{matrix} 4 & 05 & 6 \end{matrix}] \end{matrix}

(7)

The output motion of the parallel mechanism can be expressed as follows:

O M = O b_{1} \cap O b_{2} \cap O b_{3} \cap O b_{4} = [\begin{matrix} 01 & 02 & 3 & 0 & 0 & 6 \end{matrix}]

(8)

For 4PPRRR, PRRR-3PPRRR, and RRR-PRRR-2PPRRR parallel mechanism, the analysis results can also be obtained, as shown in Table 2, where 01 and 02 correspond to the linear ancillary motion along the x-axes and y-axes directions, respectively, and the local constraint helix

$_{3}^{r}, $_{4}^{r}

also correspond to the linear motion along the x-axes and y-axes directions, respectively. So ancillary motion corresponds exactly to the local constraint helix motion.

Table 2.

Comparison of different DOF analysis methods for the same mechanism.

Mechanism	Inverse helix theory (A)	Modified G-K formula (B)	MV-DOF method(C)	Analysis
4PPRRR	$$_{m 1}^{r r} = (001; 000)$ $$_{m 2}^{r r} = (000; 100)$ $$_{m 3}^{r r} = (000; 010)$ $$_{m 4}^{r r} = (000; 001)$	$M = 6 - 0 (18 - 20 - 1) + 20 + 2 = 4$	$O M = O b_{1} \cap O b_{2} \cap O b_{3} \cap O b_{4}$ $O M = [\begin{matrix} 01 & 02 & 3 \end{matrix} \begin{matrix} 0 & 0 & 6 \end{matrix}]$	A = B = C
4PPRRR	$$_{1}^{r} = (000; 100)$ $$_{2}^{r} = (000; 010)$	$μ = 2$	01, 02	Ancillary motion is equivalent to the local constraint helix motion
PRRR-3PPRRR	$$_{m 1}^{r r} = (001; 000)$ $$_{m 2}^{r r} = (000; 100)$ $$_{m 3}^{r r} = (000; 001)$	$M = 6 - 0 (17 - 19 - 1) + 19 + 2 = 3$	$O M = O b_{1} \cap O b_{2} \cap O b_{3} \cap O b_{4}$ $O M = [\begin{matrix} 01 & 02 & 3 \end{matrix} \begin{matrix} 0 & 0 & 6 \end{matrix}]$	$A \neq B \neq C$
PRRR-3PPRRR	$$_{3}^{r} = (000; 100)$ $$_{4}^{r} = (000; 010)$	$μ = 2$	01, 02	Ancillary motion is equivalent to the local constraint helix motion
RRR-PRRR-2PPRRR	$$_{m 1}^{r r} = (001; 000)$ $$_{m 2}^{r r} = (000; 001)$	$M = 6 - 0 (15 - 17 - 1) + 17 + 2 = 1$	$O M = O b_{1} \cap O b_{2} \cap O b_{3} \cap O b_{4}$ $O M = [\begin{matrix} 01 & 02 & 0 \end{matrix} \begin{matrix} 0 & 0 & 6 \end{matrix}]$	$A \neq B \neq C$
RRR-PRRR-2PPRRR	$$_{3}^{r} = (000; 100)$ $$_{4}^{r} = (000; 010)$	$μ = 2$	01, 02	Ancillary motion is equivalent to the local constraint helix motion
2RRR-PRRR-PPRRR	$$_{m 1}^{r r} = (001; 000)$ $$_{m 2}^{r r} = (000; 001)$	$M = 6 - 0 (13 - 15 - 1) + 15 + 3 = 0$	$O M = O b_{1} \cap O b_{2} \cap O b_{3} \cap O b_{4}$ $O M = [\begin{matrix} 01 & 02 & 0 \end{matrix} \begin{matrix} 0 & 0 & 6 \end{matrix}]$	$A \neq B \neq C$
2RRR-PRRR-PPRRR	$$_{31}^{r} = (010; 000)$ $$_{32}^{r} = (000; 100)$ $$_{4}^{r} = (000; 010)$	$μ = 3$	01, 02	Ancillary motion is equivalent to partial local constraint helix motion
2RRR-2PRRR	$$_{m 1}^{r r} = (001; 000)$ $$_{m 2}^{r r} = (000; 001)$	$M = 6 - 0 (12 - 14 - 1) + 14 + 4 = 0$	$O M = O b_{1} \cap O b_{2} \cap O b_{3} \cap O b_{4}$ $O M = [\begin{matrix} 01 & 02 & 0 \end{matrix} \begin{matrix} 0 & 0 & 6 \end{matrix}]$	$A \neq B \neq C$
2RRR-2PRRR	$$_{31}^{r} = (010; 000)$ $$_{32}^{r} = (000; 100)$ $$_{41}^{r} = (100; 000)$ $$_{42}^{r} = (000; 010)$	$μ = 4$	01, 02	Ancillary motion is equivalent to partial local constraint helix motion
4RRR	$$_{m 1}^{r r} = (001; 000)$ $$_{m 2}^{r r} = (000; 001)$	$M = 6 - 0 (10 - 12 - 1) + 12 + 4 = - 2$	$O M = O b_{1} \cap O b_{2} \cap O b_{3} \cap O b_{4}$ $O M = [\begin{matrix} 01 & 02 & 0 \end{matrix} \begin{matrix} 0 & 0 & 6 \end{matrix}]$	$A \neq B \neq C$
4RRR	$$_{31}^{r} = (010; 000)$ $$_{32}^{r} = (000; 100)$ $$_{41}^{r} = (100; 000)$ $$_{42}^{r} = (000; 010)$	$μ = 4$	01, 02	Ancillary motion is equivalent to partial local constraint helix motion

Similarly, for the 2RRR-PRRR-PPRRR parallel mechanism, 01 and 02 correspond to the linear ancillary motion along the x-axes and y-axes directions, respectively. The local constraint helix $$_{32}^{r}, $_{42}^{r}$ correspond to the linear motion along the spatial axes x and y directions, and $$_{41}^{r}, $_{31}^{r}$ correspond to the rotational motion around x-axes and y-axes directions. It is obvious that the output ancillary motion does not include the rotational motion, so the output ancillary motion of the branch chain does not always correspond exactly to the local restraint helix motion. Similarly, we can found that the output motion of the 2RRR-2PRRR and 4RRR parallel mechanism also have the above characteristics. In summary, the ancillary motion $(\sum V^{'})$ is equivalent to the local constraint helix motion $(\sum $^{r})$ , while in turn the local constraint helix motion is not exactly the ancillary motion, that is, $\sum V^{'} \in \sum $^{r}$ .

The same branch chain in the same mechanism is analyzed by the above method, similar features are also found. Taking the 4PPRRR mechanism as an example, the output ancillary motion on branch chains 1 and 3 are both linear motion along the x-axes and y-axes directions and rotational motion around the x-axes, while the local restraint helix motion on branch chains 1 and 3 correspond to the linear motion along the x-axes. The output ancillary motion on branch chains 2 and 4 is linear motion along the x-axis and y-axis directions and rotational motion around the y-axis, while the local restraint helix on branch chains 2 and 4 corresponds to the linear motion along the y-axis. Similarly, for other branch chains of the PRRR-3PPRRR, RRR-PRRR-2PPRRR, 2RRR-PRRR-PPRRR, 2RRR-2PRRR, and 4RRR parallel mechanism, there are also cases in which the output ancillary motion of the branch chain does not correspond exactly to the local restraint helix motion.

Careful analysis of the DOF results obtained by different methods in Table 2, it can be seen that sometimes A is equal to B, sometimes it is not, and even sometimes B is clearly wrong. The calculation results of C show good correspondence to the process of metamorphic evolution of the parallel mechanism example. Therefore, compared with traditional inverse helix theory (A) and modified G-K formula (B), the MV-DOF method (C) can always correspond to the transformation and evolution process of the parallel mechanism, and accurately analyze the DOF of the mechanism regardless of the change of the branch chains. In addition, it can also reflect the mapping relationship between input motion and output motion. The output primary motion elements are obtained by independent transfer mapping of the input motion elements, whereas the output ancillary motion elements are obtained by non-independent transfer mapping of the input motion elements.

DOF cutting analysis of mechanism based on the MV-DOF method

DOF of the mechanism is a measure of the output motion capability characteristics of the mechanism. Its essence is the mathematical representation of the possible motion forms of space linear motion and rotational motion at the output point. In order to conveniently analyze the DOF of an output point on the mechanism, especially for the complex mechanism, the output reference point can be used as the end point, along which the mechanism can be cut into two independent parts. These two parts do not affect each other and can be used as two independent sub-mechanisms. This MV-DOF method presented in this research can be used to analyze the output motion capability of each sub-mechanism independently. Finally, the intersection is obtained and used as the DOF of the whole mechanism at this output point. In special cases, the DOF of an output point on the simple mechanism can be obtained by simply observation.

DOF cutting analysis of the plane mechanism

For the selected cut at the output reference point of the plane mechanism, the cut is marked as a red line segment and labeled as $J_{i}$ , two joint symbols can also be joined with a symbol such as $J_{i - j}$ , where the values of i and j are joint mark numbers 1, 2, 3, …. Starting from the fixed joint point, the cutting branch formed in the clockwise direction is denoted as $\vec{J_{i}}$ and that formed in the counter-clockwise direction is denoted as $\overset{\leftarrow}{J_{i}}$ . The input motion matrix, the output primary motion matrix, the output first-order ancillary motion matrix, and the output motion matrix corresponding to the two cutting branches are $I (\vec{J_{i}}), O M (\vec{J_{i}}), O P_{1} (\vec{J_{i}}), O P_{2} (\vec{J_{i}}), O P_{3} (\vec{J_{i}}), O (\vec{J_{i}})$ and $I (\overset{\leftarrow}{J_{i}}), O M (\overset{\leftarrow}{J_{i}}), O P_{1} (\overset{\leftarrow}{J_{i}}), O P_{2} (\overset{\leftarrow}{J_{i}}), O P_{3} (\overset{\leftarrow}{J_{i}}), O (\overset{\leftarrow}{J_{i}})$ , respectively. The output motion matrix of the cutting reference point can be expressed as follows:

O (J_{i}) = O (\vec{J_{i}}) \cap O (\overset{\leftarrow}{J_{i}})

(9)

DOF cutting analysis is applied for a parallelogram mechanism in Figure 2 and a common quadrilateral mechanism in Figure 3. The direction perpendicular to the paper surface is defined as the z-axis direction, it can be seen by simple observation that

\begin{aligned} O (J_{1}) & = O (\vec{J_{1}}) \cap O (\overset{\leftarrow}{J_{1}}) \\ = [0 0 0 0 0 6] \cap [01 02 0 0 0 6] \\ = [0 0 0 0 0 6] \end{aligned}

(10)

\begin{aligned} O (J_{2}) & = O (\vec{J_{2}}) \cap O (\overset{\leftarrow}{J_{2}}) \\ = [01 02 0 0 0 6] \cap [01 02 0 0 0 6] \\ = [01 02 0 0 0 6] \end{aligned}

(11)

\begin{aligned} O (J_{3}) & = O (\vec{J_{3}}) \cap O (\overset{\leftarrow}{J_{3}}) \\ = [01 02 0 0 0 6] \cap [0 0 0 0 0 6] \\ = [0 0 0 0 0 6] \end{aligned}

(12)

Figure 2.

Sketch of the degree of freedom (DOF) cutting analysis of a parallelogram mechanism with fixed rods.

Figure 3.

Sketch of the degree of freedom (DOF) cutting analysis of a common quadrilateral mechanism with fixed rods.

Similarly, for the common hexagon mechanism shown in Figure 4, it can also be observed that

\begin{aligned} O (J_{1}) & = O (\vec{J_{1}}) \cap O (\overset{\leftarrow}{J_{1}}) \\ = [0 0 0 0 0 6] \cap [01 02 0 0 0 6] \\ = [0 0 0 0 0 6] \end{aligned}

(13)

\begin{aligned} O (J_{2}) & = O (\vec{J_{2}}) \cap O (\overset{\leftarrow}{J_{2}}) \\ = [01 02 0 0 0 6] \cap [01 02 0 0 0 6] \\ = [01 02 0 0 0 6] \end{aligned}

(14)

\begin{aligned} O (J_{3}) & = O (\vec{J_{3}}) \cap O (\overset{\leftarrow}{J_{3}}) \\ = [01 02 0 0 0 6] \cap [01 02 0 0 0 6] \\ = [01 02 0 0 0 6] \end{aligned}

(15)

\begin{aligned} O (J_{4}) & = O (\vec{J_{4}}) \cap O (\overset{\leftarrow}{J_{4}}) \\ = [01 02 0 0 0 6] \cap [01 02 0 0 0 6] \\ = [01 02 0 0 0 6] \end{aligned}

(16)

\begin{aligned} O (J_{5}) & = O (\vec{J_{5}}) \cap O (\overset{\leftarrow}{J_{5}}) \\ = [01 02 0 0 0 6] \cap [0 0 0 0 0 6] \\ = [0 0 0 0 0 6] \end{aligned}

(17)

Figure 4.

Sketch of the degree of freedom (DOF) cutting analysis of a common hexagon mechanism with fixed rods.

From the above analysis, it can be concluded that there are two possible forms of DOF output for the mechanism shown in Figures 2 to 4. One output form is one rotational motion around the z-axis, such as $J_{1}$ and $J_{5}$ , the other output form is one rotational motion around the z-axis and two linear motions along the x-axes and y-axes, respectively, such as $J_{2}, J_{3},$ and $J_{4}$ .

DOF cutting analysis of space mechanism

For a space mechanism in Figure 5, the space coordinate system is constructed as shown. The output motion at the output point E will be analyzed. $J_{6}$ is selected as the cutting point, which can be extended to point E. It can be observed that

O (\overset{\leftarrow}{J_{6}}) = [1 2 3 04 05 06]

(18)

Figure 5.

Sketch of the degree of freedom (DOF) cutting analysis of a space mechanism.

Considering the inclusion relationship $(\overset{\leftarrow}{J_{6}} \subset \vec{J_{6}})$ exists between the cutting branch chains, the results of DOF analysis at the output reference point E of the mechanism can be expressed as follows:

\begin{aligned} O (E) & = O (J_{6}) \\ = O (\vec{J_{6}}) \cap O (\overset{\leftarrow}{J_{6}}) \\ = O (\overset{\leftarrow}{J_{6}}) \\ = [1 2 3 04 05 06] \end{aligned}

(19)

From the above analysis, it can be obtained that, there are three linear output motions along the x-axes, y-axes, and z-axes, respectively, and three rotational output motions around the x-axes, y-axes, and z-axes, respectively, at the output point E for this space mechanism.

Conclusions

In this article, based on the primary-ancillary motion theory and its six-element characteristic numerical method, a new DOF analysis method (MV-DOF method) of parallel mechanism is constructed, and the following conclusions are obtained through example analysis and verification:

There are inaccuracies and inapplicability in DOF analysis of the 4PPRRR series-parallel mechanism by using traditional inverse helix theory and modified G-K formula. The MV-DOF method proposed in this paper shows higher accuracy and applicability.

The ancillary motion is the motion constrained off by the local constraint helix, but the motion corresponding to the local constraint helix is not exactly the ancillary motion.

The output reference point on the mechanism can be taken as the cutting point, along which the mechanism can be cut into two independent sub-mechanisms. The output DOF of each sub-mechanism can be analyzed independently by using the presented MV-DOF method, then the intersection can be taken as the DOF of the whole mechanism at this output reference point.

Compared with the traditional inverse helix theory and the modified G-K formula, specific symbols, and operation rules are defined in this MV-DOF method, the primary motion and ancillary motion of the mechanism are distinguished, and the mapping relationship between input motion and output motion is clearly reflected. Higher accuracy and better applicability are shown in analyzing the parallel mechanism example. This research solves the DOF analysis which cannot be accurately analyzed by traditional methods, and provides a new and effective DOF analytical way for parallel mechanism.

Footnotes

Acknowledgements

The authors would like to give their acknowledgment to the Natural Science Foundation of Hebei Province of China (No. E2019508105), the Central Guiding Local Technology Development Foundation of Hebei Province of China (No. 246Z3706G), and the Science and Technology Research Foundation for Higher Education Institutions of Hebei Province of China (QN2017410) for the financial support on this article.

Additional declarations

The authors declare that all the figures/tables are created by the authors of this article.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the Science and Technology Research Project of Higher Education Institutions in Hebei Province, Natural Science Foundation of Hebei Province of China (grant numbers QN2017410 and E2019508105).

ORCID iD

Min Zhang

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Research on a new method for the degree of freedom analysis of parallel mechanism

Abstract

Keywords

Introduction

Primary-ancillary motion

Comparison of different DOF analysis methods for parallel mechanism

DOF cutting analysis of mechanism based on the MV-DOF method

DOF cutting analysis of the plane mechanism

DOF cutting analysis of space mechanism

Conclusions

Footnotes

Acknowledgements

Additional declarations

Declaration of conflicting interests

Funding

ORCID iD

References